So let's think a little bit about powers of 0.
So what do you think 0 to the first power is going to be?
And I encourage you to pause this video.
Well, let's just think about it.
One definition of exponentiation is that you start with a 1,
and then, you multiply this number times a 1 one time.
So this is literally going to be 1 times--
let me do it in the right color-- it's 1 times 0.
You're multiplying the 1 by 0 one time.
1 times 0, well, that's just going to be equal to 0.
Now, what do you think 0 squared or 0 to the second power
is going to be equal to?
Well, once again, one way of thinking about this
is that you start with a 1, and we're
going to multiply it by 0 two times.
So times 0 times 0.
Well, what's that going to be?
Well, you multiply anything times 0,
once again, you are going to get 0.
And I think you see a pattern here.
If I take 0 to any non-zero number--
so to the power of any non-zero, so this
is some non-zero number, then this is going to be equal to 0.
Now, this raises a very interesting question.
What happens at 0 to the 0-th power?
So here, 0 to the millionth power is going to be 0.
0 to the trillionth power is going to be 0.
Even negative or fractional exponents,
which we haven't talked about yet,
as long as they're non-zero, this is just
going to be equal to 0, kind of makes sense.
But now, let's think about what 0 to the 0-th power
is, because this is actually a fairly deep question.
And I'll give you a hint.
Well, actually, why don't you pause the video
and think a little bit about what 0 to the 0-th power
should be.
Well, there's two trains of thought here.
You could say, look, 0 to some non-zero number is 0.
So why don't we just extend this to all numbers
and say 0 to any number should be 0.
And so maybe you should say that 0 to the 0-th power is 0.
But then, there was another train of logic
that we've already learned, that any non-zero number,
if you take any non-zero number, and you raise it
to the 0-th power.
We've already established that you start with a 1,
and you multiply it times that non-zero number 0 times.
So this is always going to be equal to 1
for non-zero numbers.
So maybe say, hey, maybe we should extend this
to all numbers, including 0.
So maybe 0 to the 0-th power should be 1.
So we could make the argument that 0 to the 0-th power
should be equal to 1.
So you see a conundrum here, and there's actually
really good cases, and you can get actually fairly
sophisticated with your mathematics.
And there's really good cases for both of these,
for 0 to 0-th being 0, and 0 to the 0-th power being 1.
And so when mathematicians get into this situation, where
they say, well, there's good cases for either.
There's not a completely natural one.
Either of these definitions would
lead to difficulties in mathematics.
And so what mathematicians have decided to do is,
for the most part-- and you'll find people who will dispute
this; people will say, no, I like one more than the other--
but for the most part, this is left undefined.
0 to the 0-th is not defined by at least just kind
of more conventional mathematics.
In some use cases, it might be defined
to be one of these two things.
So 0 to any non-zero number, you're going to get 0.
Any non-zero number to the 0-th power, you're going to get 1.
But 0 to the 0, that's a little bit of a question mark.